CUBES
AND
CUBE ROOTS
109
CHAPTER
Cubes and Cube Roots
7
7.1 Introduction
This is a story about one of India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
Hardy – Ramanujan
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed
Number
interesting. He said it is the smallest number that can be expressed
1729 is the smallest Hardy–
as a sum of two cubes in two different ways:
Ramanujan Number. There
1729 = 1728 + 1 = 123 + 13
are an infinitely many such
1729 = 1000 + 729 = 103 + 93
numbers. Few are 4104
(2, 16; 9, 15), 13832 (18, 20;
1729 has since been known as the Hardy – Ramanujan Number,
2, 24), Check it with the
even though this feature of 1729 was known more than 300 years
numbers given in the brackets.
before Ramanujan.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.
7.2 Cubes
You know that the word ‘cube’ is used in geometry. A cube is
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...
These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
itself three times.
Figures which have
3-dimensions are known as
solid figures.
110 MATHEMATICS
We note that 1 = 1 × 1 × 1 = 13; 8 = 2 × 2 × 2 = 23; 27 = 3 × 3 × 3 = 33.
Since 53 = 5 × 5 × 5 = 125, therefore 125 is a cube number.
Is 9 a cube number? No, as 9 = 3 × 3 and there is no natural number which multiplied
by itself three times gives 9. We can see also that 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27. This
shows that 9 is not a perfect cube.
The following are the cubes of numbers from 1 to 10.
Table 1
The numbers 729, 1000, 1728
are also perfect cubes.
Number
Cube
1
2
3
4
5
6
7
8
13 = 1
23 = 8
33 = 27
43 = 64
53 = ____
63 = ____
73 = ____
83 = ____
9
93 = ____
10
103 = ____
Complete it.
There are only ten perfect cubes from 1 to 1000. (Check this). How many perfect
cubes are there from 1 to 100?
Observe the cubes of even numbers. Are they all even? What can you say about the
cubes of odd numbers?
Following are the cubes of the numbers from 11 to 20.
Table 2
We are even, so
are our cubes
We are odd so are
our cubes
Number
Cube
11
12
13
14
15
16
17
18
19
20
1331
1728
2197
2744
3375
4096
4913
5832
6859
8000
CUBES
AND
CUBE ROOTS 111
Consider a few numbers having 1 as the one’s digit (or unit’s). Find the cube of each
of them. What can you say about the one’s digit of the cube of a number having 1 as the
one’s digit?
Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc.
TRY THESE
Find the one’s digit of the cube of each of the following numbers.
(i) 3331
(ii) 8888
(iii) 149
(v) 1024
(vi) 77
(vii) 5022
(iv) 1005
(viii) 53
7.2.1 Some interesting patterns
1. Adding consecutive odd numbers
Observe the following pattern of sums of odd numbers.
1 =
1 = 13
3 +
5 =
8 = 23
7 +
9 + 11 = 27 = 33
13 + 15 + 17 + 19 = 64 = 43
21 + 23 + 25 + 27 + 29 = 125 = 53
Is it not interesting? How many consecutive odd numbers will be needed to obtain
the sum as 103?
TRY THESE
Express the following numbers as the sum of odd numbers using the above pattern?
(b) 83
(c) 73
(a) 63
Consider the following pattern.
23 – 13 = 1 + 2 × 1 × 3
33 – 23 = 1 + 3 × 2 × 3
43 – 33 = 1 + 4 × 3 × 3
Using the above pattern, find the value of the following.
(i) 73 – 63
(ii) 123 – 113
(iii) 203 – 193
(iv) 513 – 503
2. Cubes and their prime factors
Consider the following prime factorisation of the numbers and their cubes.
Prime factorisation
of a number
4=2×2
6=2×3
15 = 3 × 5
12 = 2 × 2 × 3
Prime factorisation
of its cube
each prime factor
appears three times
in its cubes
43 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23
63 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33
153 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53
123 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 23 × 23 × 33
112 MATHEMATICS
2
2
2
3
3
3
216
108
54
27
9
3
1
Observe that each prime factor of a number appears
three times in the prime factorisation of its cube.
Do you remember that
am × bm = (a × b)m
In the prime factorisation of any number, if each factor
appears three times, then, is the number a perfect cube?
Think about it. Is 216 a perfect cube?
By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3
Each factor appears 3 times. 216 = 23 × 33 = (2 × 3)3
factors can be
= 63 which is a perfect cube!
grouped in triples
Is 729 a perfect cube?
729 = 3 × 3 × 3 × 3 × 3 × 3
Yes, 729 is a perfect cube.
Now let us check for 500.
Prime factorisation of 500 is 2 × 2 × 5 × 5 × 5.
So, 500 is not a perfect cube.
Example 1: Is 243 a perfect cube?
Solution: 243 = 3 × 3 × 3 × 3 × 3
There are three
5’s in the product but
only two 2’s.
In the above factorisation 3 × 3 remains after grouping the 3’s in triplets. Therefore, 243 is
not a perfect cube.
TRY THESE
Which of the following are perfect cubes?
1. 400
2. 3375
5. 9000
6. 6859
3. 8000
7. 2025
4. 15625
8. 10648
7.2.2 Smallest multiple that is a perfect cube
Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm,
30 cm, 15 cm respectively.
Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?
Raj said, Volume of cuboid is 15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5
=2×3×3×3×5×5×5
Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make
it a perfect cube. Therefore, we need 4 such cuboids to make a cube.
Example 2: Is 392 a perfect cube? If not, find the smallest natural number by which
392 must be multiplied so that the product is a perfect cube.
Solution: 392 = 2 × 2 × 2 × 7 × 7
The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect
cube. To make its a cube, we need one more 7. In that case
392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744
which is a perfect cube.
CUBES
AND
CUBE ROOTS 113
Hence the smallest natural number by which 392 should be multiplied to make a perfect
cube is 7.
Example 3: Is 53240 a perfect cube? If not, then by which smallest natural number
should 53240 be divided so that the quotient is a perfect cube?
Solution: 53240 = 2 × 2 × 2 × 11 × 11 × 11 × 5
The prime factor 5 does not appear in a group of three. So, 53240 is not a perfect cube.
In the factorisation 5 appears only one time. If we divide the number by 5, then the prime
factorisation of the quotient will not contain 5.
53240 ÷ 5 = 2 × 2 × 2 × 11 × 11 × 11
Hence the smallest number by which 53240 should be divided to make it a perfect
cube is 5.
The perfect cube in that case is = 10648.
So,
Example 4: Is 1188 a perfect cube? If not, by which smallest natural number should
1188 be divided so that the quotient is a perfect cube?
Solution: 1188 = 2 × 2 × 3 × 3 × 3 × 11
The primes 2 and 11 do not appear in groups of three. So, 1188 is not a perfect cube. In
the factorisation of 1188 the prime 2 appears only two times and the prime 11 appears
once. So, if we divide 1188 by 2 × 2 × 11 = 44, then the prime factorisation of the
quotient will not contain 2 and 11.
Hence the smallest natural number by which 1188 should be divided to make it a
perfect cube is 44.
And the resulting perfect cube is 1188 ÷ 44 = 27 (=33).
Example 5: Is 68600 a perfect cube? If not, find the smallest number by which 68600
must be multiplied to get a perfect cube.
Solution: We have, 68600 = 2 × 2 × 2 × 5 × 5 × 7 × 7 × 7. In this factorisation, we
find that there is no triplet of 5.
So, 68600 is not a perfect cube. To make it a perfect cube we multiply it by 5.
Thus,
68600 × 5 = 2 × 2 × 2 × 5 × 5 × 5 × 7 × 7 × 7
= 343000, which is a perfect cube.
Observe that 343 is a perfect cube. From Example 5 we know that 343000 is also
perfect cube.
THINK, DISCUSS AND WRITE
Check which of the following are perfect cubes. (i) 2700 (ii) 16000 (iii) 64000
(iv) 900 (v) 125000 (vi) 36000 (vii) 21600 (viii) 10,000 (ix) 27000000 (x) 1000.
What pattern do you observe in these perfect cubes?
114 MATHEMATICS
EXERCISE 7.1
1. Which of the following numbers are not perfect cubes?
(i) 216
(ii) 128
(iii) 1000
(iv) 100
(v) 46656
2. Find the smallest number by which each of the following numbers must be multiplied
to obtain a perfect cube.
(i) 243
(ii) 256
(iii) 72
(iv) 675
(v) 100
3. Find the smallest number by which each of the following numbers must be divided to
obtain a perfect cube.
(i) 81
(ii) 128
(iii) 135
(iv) 192
(v) 704
4. Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such
cuboids will he need to form a cube?
7.3 Cube Roots
If the volume of a cube is 125 cm3, what would be the length of its side? To get the length
of the side of the cube, we need to know a number whose cube is 125.
Finding the square root, as you know, is the inverse operation of squaring. Similarly,
finding the cube root is the inverse operation of finding cube.
We know that 23 = 8; so we say that the cube root of 8 is 2.
We write 3 8 = 2. The symbol
denotes ‘cube-root.’
3
Consider the following:
Statement
Inference
13 = 1
3
1 =1
23 = 8
3
8 =
3
3
Statement
63 = 216
3
216 = 6
23 = 2
73 = 343
3
343 = 7
33 = 3
83 = 512
3
512 = 8
3
729 = 9
33 = 27
3
27 =
43 = 64
3
64 = 4
93 = 729
125 = 5
103 = 1000
53 = 125
3
Inference
3
1000 = 10
7.3.1 Cube root through prime factorisation method
Consider 3375. We find its cube root by prime factorisation:
3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53 = (3 × 5)3
Therefore,
cube root of 3375 =
3
Similarly, to find 3 74088 , we have,
3375 = 3 × 5 = 15
CUBES
AND
CUBE ROOTS 115
74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7 = 23 × 33 × 73 = (2 × 3 × 7)3
Therefore,
3
74088 = 2 × 3 × 7 = 42
Example 6: Find the cube root of 8000.
Solution: Prime factorisation of 8000 is 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
So,
3
8000 = 2 × 2 × 5 = 20
Example 7: Find the cube root of 13824 by prime factorisation method.
Solution:
13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 23 × 23 × 23 × 33.
Therefore,
3
13824 = 2 × 2 × 2 × 3 = 24
THINK, DISCUSS AND WRITE
State true or false: for any integer m, m2 < m3. Why?
7.3.2 Cube root of a cube number
If you know that the given number is a cube number then following method can be used.
Step 1
Take any cube number say 857375 and start making groups of three digits
starting from the right most digit of the number.
857
↓
second group
Step 2
Step 3
375
↓
first group
We can estimate the cube root of a given cube number through a step by
step process.
We get 375 and 857 as two groups of three digits each.
First group, i.e., 375 will give you the one’s (or unit’s) digit of the required
cube root.
The number 375 ends with 5. We know that 5 comes at the unit’s place of a
number only when it’s cube root ends in 5.
So, we get 5 at the unit’s place of the cube root.
Now take another group, i.e., 857.
We know that 93 = 729 and 103 = 1000. Also, 729 < 857 < 1000. We take
the one’s place, of the smaller number 729 as the ten’s place of the required
cube root. So, we get 3 857375 = 95 .
Example 8: Find the cube root of 17576 through estimation.
Solution: The given number is 17576.
Step 1
Form groups of three starting from the rightmost digit of 17576.
116 MATHEMATICS
Step 2
Step 3
17 576. In this case one group i.e., 576 has three digits whereas 17 has only
two digits.
Take 576.
The digit 6 is at its one’s place.
We take the one’s place of the required cube root as 6.
Take the other group, i.e., 17.
Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.
The smaller number among 2 and 3 is 2.
The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of
17576.
Thus, 3 17576 = 26 (Check it!)
EXERCISE 7.2
1. Find the cube root of each of the following numbers by prime factorisation method.
(i) 64
(ii) 512
(iii) 10648
(iv) 27000
(v) 15625
(vi) 13824
(vii) 110592
(viii) 46656
(ix) 175616
(x) 91125
2. State true or false.
(i) Cube of any odd number is even.
(ii) A perfect cube does not end with two zeros.
(iii) If square of a number ends with 5, then its cube ends with 25.
(iv) There is no perfect cube which ends with 8.
(v) The cube of a two digit number may be a three digit number.
(vi) The cube of a two digit number may have seven or more digits.
(vii) The cube of a single digit number may be a single digit number.
3. You are told that 1,331 is a perfect cube. Can you guess without factorisation what
is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
WHAT HAVE WE DISCUSSED?
1. Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be
expressed as sum of two cubes in two different ways.
2. Numbers obtained when a number is multiplied by itself three times are known as cube numbers.
For example 1, 8, 27, ... etc.
3. If in the prime factorisation of any number each factor appears three times, then the number is a
perfect cube.
4. The symbol
3
denotes cube root. For example 3 27 = 3 .